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%%文档的题目、作者与日期
%\author{王立庆（2021级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
%\title{金融观点下的随机分析基础}
\title{第2章复习题 - 随机积分}
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%\date{2021 年 9 月 14 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}\itemsep1em

\item  写出 Riemann 积分的定义。按定义计算 $$\int_0^1 tdt.$$ 

\vspace{5cm}

\item  写出 Riemann-Stieltjes 积分的定义。举例说明一般的随机变量的期望的计算公式
$$E(X) = \int_{-\infty}^{\infty} tdF_X(t).$$

\vspace{5cm}

\item  设 $f$ 是连续函数，$g$ 是有界变差函数，则下述 Riemann-Stieltjes 积分存在，
$$\int_0^1 f(t)dg(t).$$

\vspace{5cm}

\item  设 $f$ 是可微函数且导数有界，设 $B$ 是标准布朗运动，则下述 Riemann-Stieltjes 积分存在，
$$\int_0^1 f(t)dB_t(\omega).$$

\vspace{5cm}

\item  设 $B$ 是标准布朗运动。解释下述等式的含义
$$\int_0^t B_sdB_s = \frac{1}{2} (B_t^2-t).$$

\vspace{5cm}

\item  设 $C$ 是一个简单过程，设 $B$ 是标准布朗运动。解释下述 Ito 积分的含义
$$I_t(C)=\int_0^t C_sdB_s.$$

\vspace{5cm}
\newpage

\item  计算上述定义的 Ito 积分的数学期望，
$$E \int_0^t C_sdB_s.$$

\vspace{5cm}

\item  设 $C$ 是一个简单过程，设 $B$ 是标准布朗运动。则有下述等矩性质，
$$E\left( \int_0^t C_sdB_s \right)^2 = \int_0^t EC_s^2ds.$$

\vspace{5cm}

\item  设 $C^{(1)}$ 和 $C^{(2)}$ 是两个简单过程，证明 Ito 积分具有下述线性性质，
$$\int_0^t \left[ k_1C^{(1)}_s + k_2C^{(2)}_s \right] dB_s 
= k_1\int_0^t C^{(1)}_s dB_s + k_2 \int_0^tC^{(2)}_s  dB_s. $$

\vspace{5cm}
\newpage

\item  设 $f$ 是二阶连续可微函数。解释下述 Ito 公式，
$$ f(B_t) - f(B_s) = \int_s^t f'(B_x)dB_x + \frac{1}{2} \int_s^t f''(B_x)dx,\,\,\, s<t. $$

\vspace{7cm}

\item  使用 Ito 公式，解释下述等式成立，
$$B_t^2 - B_s^2 = 2\int_s^t B_sdB_s + \int_s^t dx. $$

\vspace{5cm}




\end{enumerate}

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\end{document}

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